PHY450H1S. Relativistic Electrodynamics Lecture 22 (Taught by Prof. Erich Poppitz). Energy Momentum Tensor.
Posted by peeterjoot on April 10, 2011
Covering section 32, section 33 of chapter 4 in the text .
Covering lecture notes pp. 169-172: spacetime translation invariance of the EM field action and the conservation of the energy-momentum tensor (170-172); properties of the energy-momentum tensor (172.1); the meaning of its components: energy.
I have no class notes for this lecture, as traffic conspired against me and I missed all but the last 5 minutes (a very frustrating drive!) Here’s my own walk through of the content that we must have covered, much of which I did as part of problem set 6 preparation.
Total derivative of the Lagrangian density.
Rather cleverly, our Professor avoided the spacetime translation arguments of the text. Inspired by an approach possible in classical mechanics to find that we have a conserved quantity derivable from a force law, he proceeds directly to taking the derivative of the Lagrangian density (see previous lecture notes for details building up to this).
I’ll proceed in exactly the same fashion.
Multiplying through by and renaming our derivative index using a delta function we have
We can now group the terms for
Knowing the end goal, a quantity that is expressed in terms of let’s raise the indexes, and any of the ‘s that are along side of those
Next, we want to get rid of the explicit vector potential dependence
Since the operator is a product of symmetric and antisymmetric tensors (or operators), the middle term is zero, and we are left with
This provides the desired conservation relationship
Unpacking the tensor
Energy term of the stress energy tensor.
The spatially indexed field tensor components are
A final bit of assembly gives us
Momentum terms of the stress energy tensor.
For the spatial components we have
So we have
It is simple to show that is symmetric
Pressure and shear terms.
Let’s now expand , starting with the diagonal terms . Because this repeated index isn’t summed over, things get slightly irregular, so it’s easier to drop the abstraction and just pick a specific , say, . Then we have
For the magnetic components above we have for example
So we have
Clearly, the other diagonal terms follow the same pattern, and we can do a cyclic permutation of coordinates to find
For the off diagonal terms, let’s pick and expand that. We have
Again, with cyclic permutation of the coordinates we have
In class these were all written in the compact notation
 L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.